5 3 4. (3 + 4 + 3 pts) Consider the matrix A -17 3 11 whose reduced row-echelon N 0 4 form is 0 1 1 (This is the same matrix as in #3.) 0 (a) Is the vector (3, 4, 1, 4) in Nul(A)? Justify your answer by an appropriate multiplication, and show your work for the multiplication. (b) Compute a spanning set for Nul(A). (c) Compute an example of a non-0 vector in Nul(A). 5. (5 pts each) For each of the following sets H, either verify that H is a subspace of V by verifying the properties (not using a theorem), or give a specific numeric example where one of the properties fails. (a) H = {(r + s, 2r + s,r - 2s) : r, s ER} and V = 3. (b) H = {(x, sin(x)) : x E R} and V = R2. 6. (+3 bonus pts each) (a) Using only the definition of linear independence, not row reduction or any theorem, show that {(-3, 1, 0, 0, 0), (3, 0, 1, 0, 0), (-4, 0, 0, 2, 1) } is linearly independent. (b) Use the invertible-matrix theorem to explain why an n x n matrix is invertible if and only if its columns form a basis of Rn